Optimal. Leaf size=125 \[ \frac{2 e^{-2 a-x (2 b-d)+c} \, _2F_1\left (1,\frac{1}{2} \left (\frac{d}{b}-2\right );\frac{d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}-\frac{7 e^{-2 a-x (2 b-d)+c}}{4 (2 b-d)}+\frac{e^{2 a+x (2 b+d)+c}}{4 (2 b+d)}+\frac{e^{c+d x}}{d} \]
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Rubi [A] time = 0.247492, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5511, 2194, 2227, 2251} \[ \frac{2 e^{-2 a-x (2 b-d)+c} \, _2F_1\left (1,\frac{1}{2} \left (\frac{d}{b}-2\right );\frac{d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}-\frac{7 e^{-2 a-x (2 b-d)+c}}{4 (2 b-d)}+\frac{e^{2 a+x (2 b+d)+c}}{4 (2 b+d)}+\frac{e^{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 5511
Rule 2194
Rule 2227
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int \left (\frac{7}{4} e^{-2 a+c-(2 b-d) x}+e^{-2 a+c-(2 b-d) x+2 (a+b x)}+\frac{1}{4} e^{-2 a+c-(2 b-d) x+4 (a+b x)}+\frac{2 e^{-2 a+c-(2 b-d) x}}{-1+e^{2 (a+b x)}}\right ) \, dx\\ &=\frac{1}{4} \int e^{-2 a+c-(2 b-d) x+4 (a+b x)} \, dx+\frac{7}{4} \int e^{-2 a+c-(2 b-d) x} \, dx+2 \int \frac{e^{-2 a+c-(2 b-d) x}}{-1+e^{2 (a+b x)}} \, dx+\int e^{-2 a+c-(2 b-d) x+2 (a+b x)} \, dx\\ &=-\frac{7 e^{-2 a+c-(2 b-d) x}}{4 (2 b-d)}+\frac{2 e^{-2 a+c-(2 b-d) x} \, _2F_1\left (1,\frac{1}{2} \left (-2+\frac{d}{b}\right );\frac{d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}+\frac{1}{4} \int e^{2 a+c+(2 b+d) x} \, dx+\int e^{c+d x} \, dx\\ &=-\frac{7 e^{-2 a+c-(2 b-d) x}}{4 (2 b-d)}+\frac{e^{c+d x}}{d}+\frac{e^{2 a+c+(2 b+d) x}}{4 (2 b+d)}+\frac{2 e^{-2 a+c-(2 b-d) x} \, _2F_1\left (1,\frac{1}{2} \left (-2+\frac{d}{b}\right );\frac{d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}\\ \end{align*}
Mathematica [A] time = 1.03722, size = 172, normalized size = 1.38 \[ -\frac{e^{c-\frac{a d}{b}} \left (2 \left (4 b^2-d^2\right ) e^{d \left (\frac{a}{b}+x\right )} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )+2 d (2 b-d) e^{\left (\frac{d}{b}+2\right ) (a+b x)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )+d e^{d \left (\frac{a}{b}+x\right )} (d \sinh (2 (a+b x))-2 b \cosh (2 (a+b x)))\right )}{8 b^2 d-2 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.15, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{3}{\rm csch} \left (bx+a\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -4 \, b \int \frac{e^{\left (d x + c\right )}}{{\left (4 \, b - d\right )} e^{\left (6 \, b x + 6 \, a\right )} - 2 \,{\left (4 \, b - d\right )} e^{\left (4 \, b x + 4 \, a\right )} +{\left (4 \, b - d\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{{\left (24 \, b^{2} d e^{c} + 14 \, b d^{2} e^{c} + d^{3} e^{c} +{\left (8 \, b^{2} d e^{c} - 6 \, b d^{2} e^{c} + d^{3} e^{c}\right )} e^{\left (6 \, b x + 6 \, a\right )} +{\left (64 \, b^{3} e^{c} - 24 \, b^{2} d e^{c} - 10 \, b d^{2} e^{c} + 3 \, d^{3} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} -{\left (64 \, b^{3} e^{c} + 40 \, b^{2} d e^{c} - 2 \, b d^{2} e^{c} - 3 \, d^{3} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{4 \,{\left ({\left (16 \, b^{3} d - 4 \, b^{2} d^{2} - 4 \, b d^{3} + d^{4}\right )} e^{\left (4 \, b x + 4 \, a\right )} -{\left (16 \, b^{3} d - 4 \, b^{2} d^{2} - 4 \, b d^{3} + d^{4}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right ) e^{\left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right ) e^{\left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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